High-order discontinuous Galerkin hydrodynamics with sub-cell shock capturing on GPUs

Abstract

Hydrodynamical numerical methods that converge with high-order hold particular promise for astrophysical studies, as they can in principle reach prescribed accuracy goals with higher computational efficiency than standard second- or third-order approaches. Here we consider the performance and accuracy benefits of Discontinuous Galerkin (DG) methods, which offer a particularly straightforward approach to reach extremely high order. Also, their computational stencil maps well to modern GPU devices, further raising the attractiveness of this approach. However, a traditional weakness of this method lies in the treatment of physical discontinuities such as shocks. We address this by invoking an artificial viscosity field to supply required dissipation where needed, and which can be augmented, if desired, with physical viscosity and thermal conductivity, yielding a high-order treatment of the Navier–Stokes equations for compressible fluids. We show that our approach results in sub-cell shock capturing ability, unlike traditional limiting schemes that tend to defeat the benefits of going to high order in DG in problems featuring many shocks. We demonstrate exponential convergence of our solver as a function of order when applied to smooth flows, such as the Kelvin–Helmholtz reference problem of Lecoanet et al. We also demonstrate excellent scalability of our GPU implementation up to hundreds of GPUs distributed on different compute nodes. In a first application to driven, subsonic turbulence, we highlight the accuracy advantages of high-order DG compared to traditional second-order accurate methods, and we stress the importance of physical viscosity for obtaining accurate velocity power spectra.